The new mirror matter theory has only a very rough framework with many of its aspects waiting to be greatly improved and further developed as a nascent research direction. In particular, its mathematical rigor and foundations have yet to be established. Relevant new mathematical tools and approaches are desired to be implemented in the new theory. Theoretical efforts in the past several decades on fundamental physics, especially on topological quantum field theory, string theory, and quantum gravity, need to be merged into the new theoretical framework under the guidance of the newly proposed first principles. Most importantly, the neutral hadron oscillation effects predicted by the new theory are ready to be experimentally tested in laboratory, and it is time for more observation and simulation works in astronomy and cosmology under the consideration of the new theory to be conducted.

As presented below, I’d like to say a few words on the future direction of the new theory to interested mathematicians and physicists.

**To Mathematicians:**

Hamilton’s Ricci flow technique is probably one of the best tools to describe the dimensional transition of the spacetime inflation principle of the new theory. It seems to be naturally dual, as a classical counterpart, to the concept of the renormalization group (RG) flow, in particular for scalar fields, in modern quantum field theory (QFT). In terms of a “dynamical” fiber bundle, RG flow describes the phase transition of quantum fields under the unextended intrinsic/fiber quantum spaces while Ricci flow depicts the picture of the dimensional transition of inflated classical spacetime. It is very tempting to find a way to incorporate the idea of Ricci flow, or geometric flows in general, into the theory of fiber bundles. Meanwhile, considering the duality between base manifold and fiber space, a new dynamical theory of differential geometry might be the right foundation for the new mirror matter theory.

Intriguingly, the new mirror matter theory seems to be demanding widest and profoundest mathematics. Feynman’s path integration formalism, as the form of quantum variational principle, is based on differential geometry with many still unsolved issues. The concept of probability amplitude and application of functional integral over quantum fields require probability theory and measure theory for better understanding. The consistent observation principle introduces constraints and symmetries in the new theory, which could be built on group theory, abstract algebra, algebraic topology, maybe even algebraic geometry and category theory. Because of non-vanishing Planck constant, quantumness requires discreteness. And the concept of probability amplitude requires randomness ( which seems to be closely related to the distribution of prime numbers). All these might also establish some connection to number theory.

The new mirror matter theory is the natural product of the three proposed first principles. Probably its mathematical foundations need an elegant unification of all these different mathematical branches, in particular, development and application of cutting-edge ideas and techniques in algebra, geometry, analysis, probability / measure, and even number theories.

**To Theoretical Physicists:**

Gravity and quantum phenomena are the duality of the same physical reality. A static single theory of everything is not the way for unification. Incorporating the three first principles of the new mirror matter theory is probably the best way to refresh the past works on fundamental physics. In particular, string theory is likely a mathematical tool for single-phased supersymmetric mirror models under the spacetime of given dimension. Theories like loop quantum gravity are probably more equipped for studies on dimensional transitions of spacetime.

Measurements are based on the classical extended spacetime and gravity is a pure classical phenomenon of classical spacetime. Under 4-d spacetime, the gauge groups and three generations of elementary particles in the Standard Model are determined by self-consistency of the new theory. Supersymmetry (SUSY) does not provide extra degrees of freedom for particles, but manifests as a broken symmetry for existent known particles: for example, gauge-SUSY between gauge bosons and matter fermions; chiral-SUSY between Higgs-like scalars as fermion condensates and neutrino-like singlets (no gauge coupling).

It is CP-symmetry (i.e., time reversal symmetry) that doubles the degrees of freedom of the particle zoo by adding anti-particles for corresponding particles. Similarly, it is mirror symmetry (i.e., orientation symmetry of spacetime manifold) that doubles the size of the particle zoo again by providing two sectors of ordinary and mirror particles. In the end, we have four almost independently separate sectors of particles, which could be understood as the four connected but not simply-connected components of the Lorentz group O(1,3) under 4-d spacetime. As such, the topological properties of quantum field theory are particularly important and often found in manifestations of new physics. Further developments in QFT would need to apply more mathematical ideas and tools like the aforementioned topological techniques – Ricci flow and RG flow.

**To Experimentalists and Astrophysicists:**

A universal oscillation effect for neutral hadrons that is predicted under the new theory is ready to be fully scrutinized in laboratory. The current laboratory technology is good enough to verify or refute the rather exact yet unique predictions of the new theory. In addition, it could also provide precise measurements of the very few model parameters. In particular, neuron lifetime anomaly under magnetic ultracold neutrons (UCN) traps of various sizes could be well detected, and the anomalous lifetime values could deviate from the nominal beta-decay value by more than 100 seconds for certain narrow traps. Resonant oscillations between ordinary and mirror neutrons (n-n’) could be observed under super-strong magnetic fields (on the order of 100 Tesla) due to the medium effect, which could manifest as a neutron loss rate as high as 25%.

For experimentalists in particle physics using accelerator facilities, oscillation effects for other neutral mesons and baryons could be detected as invisible decays. Some of the most significant candidates for invisible decays are: (K^{0}_{L}-K^{0′}_{L}) 9.9 x 10^{-6}, (K^{0}_{S}-K^{0}_{S}) 1.8 x 10^{-6}, (Λ^{0}-Λ^{0}’) 4.4×10^{-7}，etc. Further measurements of such oscillations involving heavier quarks (e.g., D^{0} and B^{0}) could be conducted in hopefully not too distant future, which may eventually help determining the ordinary-mirror mixing strengths of all quarks.

It is time to consider the new mirror matter theory in cosmological models like ΛCDM and it’ll help us quantitatively understand the evolution of early Universe and the formation of large structures. It is also time to introduce n-n’ oscillation effects in simulations of stellar evolution including supernova models, which will help us resolve many puzzling difficulties in late stages of stellar evolution and nucleosynthesis. Observation of ultrahigh energy cosmic rays may make it possible to study mirror astronomical objects. The 2nd sharper GZK cutoff of cosmic rays at even higher energies, if indeed observed and confirmed, could be used to determine the temperature of the mirror sector in the Universe. Works on gravitational lensing across the sky could be the best way to map out the distribution of mirror matter in the Universe. If we do have mirror objects (e.g., mirror planets) nearby, there is no way to observe them via the ordinary electromagnetic telescopes. But we should be able to “feel” their existence via the anomalous gravitational effects.